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221 quasicrystal is crystallographic, it is assumed that w transforms as a vector in the plane perpendicular to the direction in which the crystal is periodic, but differently if the point group is 5-, 8-, 10or 12-gonal. From the Neumann principle follow restrictions on the form of the phonon, phason and phonon-phason coupling contributions to the elastic stiffness matrix that can be determined by combining the restrictions obtained for a set of elements generating the point group of interest. For the phonon part, the restrictions obtained for the generating elements do not depend on the system to which the point group belongs. This remains true for the phason and coupling parts in the case of crystallographic point groups but, in general, breaks down for the non-crystallographic ones. The form of the symmetric 12 x 12 matrix giving the phonon, phason and phonon-phason coupling contributions to the elastic stiffness is presented in graphic notation.

Successive phase transition of a Cd–Yb 1/1 crystalline approximant under high pressure